The heat equation

The numerous examples of Chapters 2 and 3 demonstrate how the classical method of separation of variables is used to generate solutions of Laplace’s and the heat equation over specified finite domains. Utilizing the modern methods inherent in Green’s functions and Green’s Theorem, solutions of Poisson’s equation over an arbitrary finite domain are established. Can such modern methods be applied to the heat equation? Unlike Laplace’s or Poisson’s equations which are static in time, the heat equation evolves in time. Indeed, the heat equation is the prototype for what are commonly called evolution equations. Nonlinear evolution equations are examined in Chapter 6 and a special nonlinear system is detailed in Chapter 7. For now, the focus will be on the linear heat and wave equations. Instead of Green’s Theorem, one of the most powerful ideas in modern mathematics is applied: The Fourier transform. The program of study for this chapter then is to define the Fourier transform, develop its properties, apply it to the heat and wave equations, and derive analytic solutions.

It is known that, if the time variable in the heat equation is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. Also, it is known that if both variables, time and spatial, are complex, then, e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This ‘complexification’ is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve some of the basic differential equations. Here we have considered an ordinary differential equation of order two along with heat equation and one- and two-dimensional wave equations. A nonlinear ordinary differential equation of order two is also considered. The ordinary differential equation is reduced to a system of nonhomogeneous linear equations which is then solved by using the Gauss elimination method, whereas the heat equation and the one-dimensional and two-dimensional heat and wave equations are reduced to a system of ordinary differential equations. The system is then solved by the optimal four-stage three-order strong stability preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method have been tested on six examples.KeywordsOrdinary differential equationHeat equationWave equation cubic B-spline functionsModified cubic B-spline quadrature methodSystem of ordinary differential equationsGauss elimination methodRunge–Kutta fourth-order method

Solutions of heat or diffusion equations with the boundary conditions which is a dynamic random field are discussed. This kind of method can be used to obtain the description of heat equations or diffusion equations based on observed physical reality, ie ordinary differential equations, representing heat or diffusion propagation, with a boundary condition that satisfies stochastic differential equations. The heat or diffusion equations obtained from the method are the compared to the heat equation or the stochastic diffusion. The comparison is emphasized on the existence and properties of Green functions.

In this paper, a thermodynamically consistent phase-field model is employed to simulate the thermocapillary migration of a droplet. The model equations consist of a general Navier–Stokes equation for the two-phase flows, a Cahn–Hilliard equation for the diffuse interface, and a heat equation, and meanwhile satisfy the balance laws of mass, energy and entropy. In particular, the total energy of the system includes kinetic energy, potential energy and internal energy, which leads to a highly coupled and nonlinear equation system. We therefore develop a linear mass and energy conserving, semi-decoupled numerical method for the numerical simulations. As the model contains a heat (energy) equation, a simple error term introduced by the temporal discretization of the momentum equation can be absorbed into the heat equation, such that the numerical solutions satisfy the conservation laws of mass and energy exactly at the temporal discrete level. Several numerical tests are carried out to validate our numerical method.

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.

Radiative frost is one of the most severe weather conditions that affects agricultural activities in many parts of the world. Since various protective methods to reduce frost impact are available, refinements of frost forecasting methodologies should provide economical benefits. In the present study, a three-dimensional numerical local-scale model for the simulation of the microclimate near the ground surface of nonhomogeneous regions during radiative frost events was developed. The model is based on the equations of motion, heat, humidity and continuity in the atmosphere and the equations of heat and moisture diffusion in the soil. Emphasis was given in establishing a refined formulation of energy budget equations for soil surface and plant canopy Additionally, an improved finite difference scheme procedure for approximating horizontal derivatives in a terrain-following coordinate system was introduced. The sensitivity of the model to various parameters that way affect the nocturnal minimum temperature near ground surface during radiative frost events was tested by using one- and two-dimensional versions of the model. This temperature was found to be sensitive to topography, plant cover, soil moisture content, air specific humidity and wind velocity.

The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation by considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equation and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. The first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.

Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

This work focuses on thermal problems, solvable using the heat equation. The fundamental question being answered here is: what are the limits of the dimensions that will allow a 3-D thermal problem to be accurately modelled using a 2-D Heat Equation? The presented work solves 2-D and 3-D heat equations using the Finite Difference Method, also known as the Forward-Time Central-Space (FTCS) method, in MATLAB®. For this study, a cuboidal shape domain with a square cross-section is assumed. The boundary conditions are set such that there is a constant temperature at its center and outside its boundaries. The 2-D and 3-D heat equations are solved in a time dimension to develop a steady state temperature profile. The method is tested for its stability using the Courant-Friedrichs-Lewy (CFL) criteria. The results are compared by varying the thickness of the 3-D domain. The maximum error is calculated, and recommendations are given on the applicability of the 2-D heat equation.

A closed-form solution for temperature and stress fields is presented for short-pulse laser heating of the metal surface. Thermo-mechanical coupling between the heat and stress equations is incorporated in the analysis. The lattice site heat equation based on the non-equilibrium energy transport is used to account for the thermal field due to short-pulse heating. The Lie symmetry method is adopted to obtain the solution for the heat equation with the appropriate boundary conditions. In the analysis, stress wave dissipation is omitted in space due to mathematical simplifications. It is found that thermal displacement is negative in the early heating period and it becomes positive as the heating period progresses, which is attributed to thermo-mechanical coupling.

A class of new stable, explicit methods to solve the non‐stationary heat equation

Composite materials are becoming more popular in technological applications due to the significant weight savings and strength offered by these materials compared to metallic materials. In many of these practical situations, the structures suffer from drop-impact loads. Materials and structures significantly change their behavior when submitted to impact loading conditions compared to quasi-static loading. The present work is devoted to investigating the thermal process in carbon-fiber-reinforced polymers (CFRP) subjected to a drop test. A novel drop-weight impact test experiment is performed to evaluate parameters specific to 3D composite materials. A strain gauge rosette and infrared thermography are employed to record the kinematic and thermal fields on the composites’ surfaces. This technique is nondestructive and offers an extensive full-field investigation of a material’s response. The combination of strain and infrared thermography data allows a comprehensive analysis of thermoelastic effects in CFRP when subjected to impacts. The experimental results are validated using numerical analysis by developing a MATLAB® code to analyze whether the coupled heat and wave equation phenomenon exists in a two-dimensional polar coordinate system by discretizing through a forward-time central-space (FTCS) finite-difference method (FDM). The results show the coupling has no significant impact as the waves generated due to impact disappears in 0.015 s. In contrast, heat diffusion happens for over a one-second period. This study demonstrates that the heat equation alone governs the CFRP heat flow process, and the thermoelastic effect is negligible for the specific drop-weight impact load.

This paper employs the weighted energy method to derive estimates for the dynamic behavior of solutions to boundary and initial boundary value problems with nonhomogeneous boundary conditions. In particular, the method is applied to the heat and Laplace equations in a bounded or unbounded region. Extensions to related equations are also studied. Similar estimates but for the spatial behavior is obtained for the heat equation and the backward in time heat equation. Results for blow-up in finite time of solutions to certain nonlinear equations are generalized to include nonhomogeneous boundary conditions, while solutions that vanish on part of the boundary are briefly discussed in the final section.